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Evolving convex curves by a generalized length-preserving flow

Laiyuan Gao, Shengliang Pan

Abstract

This paper deals with a generalized length-preserving flow for convex curves in the plane. It is shown that the flow exists globally and deforms convex curves into circles as time tends to infinity.

Evolving convex curves by a generalized length-preserving flow

Abstract

This paper deals with a generalized length-preserving flow for convex curves in the plane. It is shown that the flow exists globally and deforms convex curves into circles as time tends to infinity.

Paper Structure

This paper contains 9 sections, 17 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Global existence
  3. Short time existence
  4. Uniform bounds of the support function
  5. Extending the flow globally
  6. Higher regularity of the flow
  7. Convergence of the curvature
  8. Convergence of the evolving curve
  9. Appendix: uniform gradient estimate

Key Result

Theorem 1.1

Let $X_0$ be a smooth and convex plane curve. If $F$ satisfies the above conditions (i)-(iii) then the flow (eq:1.1.201811) exists on time interval $[0, +\infty)$, keeps the convexity of the evolving curve, preserves its length and deforms $X(\cdot, t)$ into a finite circle as time tends to infinity

Theorems & Definitions (33)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • ...and 23 more